Problem Set 4: Conceptual
Problems on insurance, hedge funds, and Value at Risk.
This problem set focuses on conceptual understanding — applying fat tail thinking to real-world situations. These questions test whether you can translate mathematical insights into practical wisdom.
There are no formulas to compute here. Instead, reason carefully about the implications of fat tails for insurance, finance, and risk management.
Problem 8: Insurance and Fat Tails
Problem Statement
Explain why insurance for fat-tailed risks (e.g., pandemics) is fundamentally different from insurance for thin-tailed risks (e.g., car accidents).
Points to Consider
- How well does historical data predict future claims?
- Can premiums be priced reliably?
- What is the relationship between collected premiums and potential claims?
- What role do the Law of Large Numbers and CLT play in insurance pricing?
The Insurance Model
Traditional insurance relies on the Law of Large Numbers: pool many independent risks, and the average claim becomes predictable. But this requires:
- Independence (or weak dependence) between claims
- Finite variance of individual claims
- Historical data that is representative of future risks
Fat-tailed risks violate all three assumptions.
Thin-Tailed: Car Accidents
Car accidents are largely independent across policyholders. The distribution of claim sizes has finite variance. Decades of data provide stable estimates. An insurer can confidently set premiums to cover expected claims plus a margin.
Fat-Tailed: Pandemics
A pandemic affects all policyholders simultaneously (no independence). The severity is unbounded (could be 10x or 100x a "normal" event). Historical data is sparse and misleading — the worst pandemic in a century may not represent what is possible. A single event can exceed all premiums ever collected.
Problem 9: The Skeptical Investor
Problem Statement
A hedge fund has 10 years of returns with low volatility and a Sharpe ratio of 2. Why should you be skeptical?
Background
The Sharpe ratio measures risk-adjusted returns:
A Sharpe ratio of 2 is exceptional — most successful funds achieve 0.5-1.0.
Points to Consider
- In Extremistan, what does 10 years of data tell you?
- What kinds of strategies generate high Sharpe ratios?
- What is "volatility" actually measuring?
- What risks might be hidden in the tails?
Hidden Tail Risk
Many strategies that show stable returns are implicitly selling tail risk — collecting small premiums most of the time but facing catastrophic losses occasionally. Ten years might simply mean the tail event has not occurred yet.
The Turkey Problem
Consider Taleb's turkey: fed every day for 1000 days, with increasingly confident estimates of the farmer's benevolence. Then comes Thanksgiving. A track record in Extremistan is evidence of survival, not evidence of safety.
Why Sharpe Ratio Fails
- Uses volatility (standard deviation) as the risk measure
- Volatility only captures typical variations, not tail risk
- If returns are fat-tailed, the denominator underestimates true risk
- The ratio becomes meaningless — or worse, misleading
Problem 10: Value at Risk
Problem Statement
Why does Taleb argue that "Value at Risk" (VaR) is dangerous for fat-tailed risks?
What is VaR?
Value at Risk answers: "What is the maximum loss at the 99th percentile?"
Where is the inverse CDF (quantile function) of the loss distribution. VaR tells you: "99% of the time, losses will not exceed this amount."
The Fatal Flaw
VaR tells you the boundary of the 99th percentile, but nothing about what lies beyond. In thin-tailed distributions, the losses beyond VaR are bounded and manageable. In fat-tailed distributions, the losses beyond VaR can be arbitrarily large — potentially infinite in expectation!
The Problem Illustrated
| Distribution | VaR (99%) | E[Loss | Loss > VaR] |
|---|---|---|
| Normal | 2.33σ | Finite, modest |
| Pareto (α = 2) | Some finite value | Infinite! |
Real-World Consequence
A bank using VaR might report: "Our 99% VaR is $100 million." This sounds prudent. But if losses are fat-tailed, the expected loss conditional on exceeding VaR might be $1 billion — or unbounded. VaR provides false comfort by ignoring precisely the events that matter most.
Better Alternatives
- Expected Shortfall (CVaR): Average loss given that loss exceeds VaR — at least considers tail severity
- Stress testing: Explicitly model extreme scenarios
- Tail risk measures: Focus on what happens in the tails, not just the boundary
What You Should Learn
- Insurance for fat-tailed risks cannot rely on the Law of Large Numbers — single events can exceed all collected premiums
- Track records in Extremistan are evidence of survival, not evidence of safety — hidden tail risks may not have materialized yet
- The Sharpe ratio and volatility-based risk measures are misleading under fat tails
- VaR ignores the severity of tail events — it tells you where the cliff is, not how far down it goes
- These are not academic concerns — they represent fundamental failures in how risk is measured and managed in practice